Two spheres of masses 4 kg and 8 kg are moving with velocities `2 ms^(-1) and 3 ms^(-1)` away from each other along the same line. The velocity of centre of mass is

Options

A. `(8)/(3)ms^(-1)` towards second sphere
B. `(8)/(3)ms^(-1)` towards first sphere
C. `(4)/(3)ms^(-1)` towards second sphere
D. `(4)/(3)ms^(-1)` towards first sphere

Answer

To find the velocity of the center of mass of the two spheres, we can follow these steps:

### Step 1: Identify the masses and velocities
- Mass of the first sphere, \( m_1 = 4 \, \text{kg} \)
- Velocity of the first sphere, \( v_1 = -2 \, \text{m/s} \) (negative because it is moving in the opposite direction)
- Mass of the second sphere, \( m_2 = 8 \, \text{kg} \)
- Velocity of the second sphere, \( v_2 = 3 \, \text{m/s} \) (positive because it is moving in the positive direction)

### Step 2: Write the formula for the velocity of the center of mass
The velocity of the center of mass \( V_{cm} \) is given by the formula:
\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ V_{cm} = \frac{(4 \, \text{kg})(-2 \, \text{m/s}) + (8 \, \text{kg})(3 \, \text{m/s})}{4 \, \text{kg} + 8 \, \text{kg}} \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:
\[ = (4 \times -2) + (8 \times 3) = -8 + 24 = 16 \]
Calculating the denominator:
\[ = 4 + 8 = 12 \]

### Step 5: Calculate the velocity of the center of mass
Now, substituting back into the equation:
\[ V_{cm} = \frac{16}{12} = \frac{4}{3} \, \text{m/s} \]

### Step 6: Determine the direction of the center of mass velocity
Since the result is positive, it indicates that the center of mass is moving in the direction of the second sphere (which is the positive direction).

### Final Answer:
The velocity of the center of mass is \( \frac{4}{3} \, \text{m/s} \) towards the second sphere.

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